Granite products is a significant part of the economy around here in Colorado, but outside of countertops and grave stones you really won't find much. It's difficult to work fine textures into, and when left to the elements won't retain them long, so it's ot favored for sculpture. As hard as it is, it's very porous, so when used as a pot will likely suffer significant damage from freez/thaw cycles over winter. This all makes it a poor choice for pots.
Basalt more homogeneous, so less porous and prone to weathering, but finding large enough chunks of pure basalt, without any other materials in it, to make a sizable pot is rare. Still will suffer from the moisture retention and freeze damage, though.
Marble beats almost anything in stone if you're looking for something like a pot. Less porous, fairly homogeneous, very hard. It's moisture retention is much lower than other stones, but not low enough.
All of the above options would be VERY expensive.
There's a reason ceramic pots are the standard. High fired ceramic is almost completely homogeneous, and very dense and strong. You can make a lighter pot with less material that will stand up better to weathering.
If you're going for quality that's cheap, can be made good looking, and will last long enough for a tree to earn a nicer pot, cement is your best option. Only drawbacks are with holding fine textures if sculpted, and weight, but these are similar problems you'd have with stone, only for pennies on the dollar.
